Let $T:\mathbb{R^4} \to \mathbb{R^4}$ which maps orthogonal vectors into orthogonal vectors. Is true or false that:
a) $T$ is an isomorphism?
b) $\langle u,v\rangle=\langle T(u),T(v)\rangle$ $\forall\ u,v \in \mathbb{R^4} $?
For the point a. I don't know what to do or to find a counterexample; for the second one I could find a counterexample in the case $u=v$?